The Gorenstein Colength of an Artinian Local Ring

In this paper, we make the notion of approximating an Artinian local ring by a Gorenstein Artin local ring precise using the concept of Gorenstein colength. We also answer the question as to when the Gorenstein colength is at most two.Comment: The title and layout have changed, has been shortened to 9 pages. Published in Journal of Algebr

Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type

In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay type.Comment: 15 Page

Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules \Ext_R^i(M,N) which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of \Ext_R^i(M,N) needed to ensure that \Ext_R^i(M,N)=0 for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two

The Ionizing Radiation-Induced Bystander Effect: Evidence, Mechanism, and Significance

It has long been considered that the important biological effects of ionizing radiation are a direct consequence of unrepaired or misrepaired DNA damage occurring in the irradiated cells. It was presumed that no effect would occur in cells in the population that receive no direct radiation exposure. However, in vitro evidence generated over the past two decades has indicated that non-targeted cells in irradiated cell cultures also experience significant biochemical and phenotypic changes that are often similar to those observed in the targeted cells. Further, nontargeted tissues in partial body-irradiated rodents also experienced stressful effects, including oxidative and oncogenic effects. This phenomenon, termed the “bystander response,” has been postulated to impact both the estimation of health risks of exposure to low doses/low fluences of ionizing radiation and the induction of second primary cancers following radiotherapy. Several mechanisms involving secreted soluble factors, oxidative metabolism, gap-junction intercellular communication, and DNA repair, have been proposed to regulate radiation-induced bystander effects. The latter mechanisms are major mediators of the system responses to ionizing radiation exposure, and our knowledge of the biochemical and molecular events involved in these processes is reviewed in this chapter